You may have a drainage issue. If it's a sunken area, you might be able to fill it in with a load of dirt. Just remember thea water will drain somewhere and think ahead so you don't cause a problem in another area. Another solution may be to put in a perforated drainage pipe and run it toward a storm drain or area that naturally drains well. If you get the area dried out, you may be able to plant shad grass or some shade plants.
I live in Oregon, and moss on sidewalks and walkways is a real problem. I purchase the cheapest detergent I can, usually in bulk or at the dollar tree, and sprinkle it on. In just days, it turns brown and can be lifted off the lawn or in my case, swept away. Works every time!
Replace that part of the lawn with stone in the form of round gravel or that plus stepping stones or flagstone. If conditions are not right for grass, like too dark and damp, then plant nothing or other shade plants. I would plant other shade plants, check out hellebores (google them) ferns and hostas, and do a stone path through them.
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